Chapter Seven. Chapter Seven

Transcription

1 Chapter Seven Chapter Seven

2 ConcepTests for Section 7. CHAPTER SEVEN 7. Which of the following is an antiderivative of + 7? (e) + C. The most general antiderivative is C, so is one possible antiderivative.. For t > 0 which of the following is an antiderivative of f(t) = t? ln(t) ln(t) ln(t) t (e) None of these. We can rewrite the function as f(t) = t = t, so an antiderivative is ln(t).. Which of the following is an antiderivative of g() = 6sin()? 8 cos() 8 sin() sin() cos() (e) 6 cos(). If students get this wrong, have them find the derivative of each of the options. ConcepTests for Section 7.. Which of the following can be converted to the form w 0 dw using a substitution? ( + ) 0 d ( + ) 0 d ( + ) 0 d ( + ) 0 d (e) ( + ) 0 d. We use w = +, so dw = d so we need. The onl one that can be converted to this form is.

3 8 CHAPTER SEVEN. Which of the following can be converted to the form cos + sin d + d + d + d (e) e + d and onl. dw using a substitution? w. Which of the following can be converted to the form e w dw using a substitution? e d e d e d e cos d onl. For Problems, can the integral can be converted to the form w n dw b a substitution, where n is a constant? +. + d e e (e + e ) d + d sin d,, and. In w = + ields w / dw, in w = e + e ields w dw, in w = + ields w dw.. 6 ( 7 + 6) 6 d 6 ( 7 + 6) 6 d d e e + 6 d and. In w = ields 7 w 6 dw, in w = e + 6 ields w dw. You might note wh w = does not work for choice.

4 ConcepTests for Section 7. CHAPTER SEVEN 9. b a 6 d = 6b 6a 6a 6b 6b + 6a b a (e) a b (f) b + a (g) 6(b a) + C. We find the antiderivative and use the Fundamental Theorem of Calculus.. Using a substitution, we have 0 ( + ) d = w dw (e) 0 w dw w dw w dw 0 w dw (f) 0 w dw (e). We use the substitution w = +. For the new limits of integration, notice that = 0 gives w = and = gives w =.. If 6 f(t)dt = 8, then f() d = (e) 6. We use the substitution t =. Then dt = d and when = 0 we have t = 0 and when = we have t = 6. Therefore f() d = 6 f(t) dt = 8 =. 0 0

5 0 CHAPTER SEVEN ConcepTests for Section 7.. Which of the following graphs could represent an antiderivative of the function shown in Figure 7.? 6 Figure Because the graph in Figure 7. is decreasing, the graph of the antiderivative must be concave down. Because the graph in Figure 7. is positive for <, zero for =, and negative for >, then = is a local maimum. You can mimic this problem with =.

6 CHAPTER SEVEN. Which of the following graphs could represent an antiderivative of the function shown in Figure 7.? π/ π Figure 7. π/ π π/ π π/ π π/ π. Because the graph in Figure 7. is alwas positive on this interval, the antiderivative must be increasing for this interval. You can mimic this problem with = sin. You could also ask our students to draw other possible antiderivatives of the function shown in Figure 7..

7 CHAPTER SEVEN. Which of the following graphs could represent an antiderivative of the function shown in Figure 7.? Figure and. Because the graph in Figure 7. is continuall increasing, the graph of its antiderivative is concave up. Notice that the graphs in and differ onl b a vertical shift. You could also point out that since the graph in Figure 7. is negative from < < 0, then the graph of an antiderivative must be decreasing on this interval. Also, since the graph in Figure 7. is positive for 0 < <, then the graph of an antiderivative must be increasing on this interval.

8 CHAPTER SEVEN. Consider the graph of f () in Figure 7.. Which of the functions with values from the Table 7. could represent f()? Table f () g() h() j() 6 k() Figure 7.,,, You might point out onl relative values of functions are important for this problem, not the actual values.. Graphs of the derivatives of four functions are shown in (I) (IV). For the functions (not the derivative) list in increasing order which has the greatest change in value on the interval shown. (I), (IV), (III), (II) (I), (IV), (II), (III) (I) = (II), (IV), (III) (I) = (II), (III) = (IV) (e) (I) = (II) = (III) = (IV) (I) (II) (III) (IV)

9 CHAPTER SEVEN (e). The ordering will be given b the values of the area under the derivative curves. These areas are (I) (/)()() = 8, (II) (/)()( + ) + (/)()() = 8, (III) () + () = 8, and (IV) () = 8. You can draw some other graphs of derivatives and ask which function has the greatest change over a specified interval. 6. Graphs of the derivatives of four functions are shown in (I) - (IV). For the functions (not the derivative) list in increasing order which has the greatest change in value on the interval shown. (I), (III), (IV), (II) (I) = (III), (IV), (II) (IV), (I) = (III), (II) (I) = (III) = (IV), (II) (e) (I) = (II) = (III) = (IV) (I) (II) (III) (IV). The ordering will be given b the values of the area under these curves. These areas are (I) ()+(/)()() = 0, (II) () + (/)()() =, (III) same as (I), 0, and (IV) (/)()( + ) + () = 0. You could ask students how to change graph (II) to have area equal to 0 and not be identical to an of the other graphs.

10 CHAPTER SEVEN 7. Figure 7. shows a graph of = f() with some areas labeled. Assume F () = f() and F(0) = 0. Then F() = 7 (e) (f) 6 Area = 7 f() Area = 6 Figure 7. (e). Since f() d = 7 6 =, we know that the total change in the antiderivative F() between = 0 and 0 = is +, so F() = 0 + =. 8. Figure 7.6 shows a graph of = F (). Where is there a local maimum on F()? A local minimum? 7 ; and 7 ;. ; 6 6 ; (e). ; and 7 (f) 7 ; F () Figure 7.6. Since we see that F () is positive before, negative between and 6, then positive after 6, we know that F() is increasing before, decreasing between and 6, and then increasing again after 6. Therefore, we know that F() has a local maimum at = and a local minimum at = 6.